5-Minute Check on Lesson 6-4 Transparency 6-5 Click the mouse button or press the Space Bar to display the answers. Refer to the figure 1.If QT = 5, TR = 4, and US = 6, find QU. 2.If TQ = x + 1, TR = x – 1, QU = 10 and QS = 15, find x. Refer to the figure 3.If AB = 5, ED = 8, BC = 11, and DC = x – 2, find x so that BD // AE. 4.If AB = 4, BC = 7, ED = 5, and EC = 13.75, determine whether BD // AE. 5. Find the value of x + y in the figure? Standardized Test Practice: ACBD Yes 19.6 B 3 R T Q U S A B C D E 5y – 6 2y + 3 3x – 2 2x + 1

Lesson 6-5 Parts of Similar Triangles

Objectives Recognize and use proportional relationships of corresponding perimeters of similar triangles Recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles

Vocabulary None New

Theorems If two triangles are similar then –The perimeters are proportional to the measures of corresponding sides –The measures of the corresponding altitudes are proportional to the measures of the corresponding sides –The measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides –The measures of the corresponding medians are proportional to the measures of the corresponding sides Theorem 6.11: Angle Bisector Theorem: An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides

Special Segments of Similar Triangles If ∆PMN ~ ∆PRQ, then PM PN MN special segment = = = AB AC BC special segment ratios of corresponding special segments = scaling factor (just like the sides) in similar triangles Example: PM 1 median PQ = = --- AB 3 median AD 3 Special segments are altitudes, medians, angle and perpendicular bisectors P M A N B C D Q

Similar Triangles -- Perimeters P Q R M N If ∆PMN ~ ∆PRQ, then Perimeter of ∆PMN PM PN MN = = = Perimeter of ∆PRQ PR PQ RQ ratios of perimeters = scaling factor (just like the sides)

Angle Bisector Theorem - Ratios P QRN If PN is an angle bisector of  P, then the ratio of the divided opposite side, RQ, is the same as the ratio of the sides of  P, PR and PQ PR RN = PQ NQ

If ∆ABC~∆XYZ, AC=32, AB=16, BC=16  5, and XY=24, find the perimeter of ∆XYZ Let x represent the perimeter of The perimeter of C Example 1a

Proportional Perimeter Theorem Substitution Cross products Multiply. Divide each side by 16. Answer: The perimeter of Example 1a cont

If ∆PNO~∆XQR, PN=6, XQ=20, QR=20  2, and RX = 20, find the perimeter of ∆PNO Answer: R Example 1b

∆ ABC and ∆ MNO are similar with a ratio of 1:3. (reverse of the numbers). According to Theorem 6.8, if two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides. Answer: The ratio of the lengths of the altitudes is 1:3 or ⅓ ∆ ABC ~∆ MNO and 3BC = NO. Find the ratio of the length of an altitude of ∆ABC to the length of an altitude of ∆MNO Example 2a

Answer: ∆EFG~∆MSY and 4EF = 5MS. Find the ratio of the length of a median of ∆EFG to the length of a median of ∆MSY. Example 2b

In the figure, ∆EFG~ ∆JKL, ED is an altitude of ∆EFG and JI is an altitude of ∆JKL. Find x if EF=36, ED=18, and JK=56. K Write a proportion. Cross products Divide each side by 36. Answer: Thus, JI = 28. Example 3a

Answer: 17.5 N In the figure, ∆ABD ~ ∆MNP and AC is an altitude of ∆ABD and MO is an altitude of ∆MNP. Find x if AC=5, AB=7 and MO=12.5 Example 3b

The drawing below illustrates two poles supported by wires with ∆ABC~∆GED, AF  CF, and FG  GC  DC. Find the height of the pole EC. are medians of since and If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. This leads to the proportion Example 4

measures 40 ft. Also, since both measure 20 ft. Therefore, Write a proportion. Cross products Simplify. Divide each side by 80. Answer: The height of the pole is 15 feet. Example 4 cont

5-Minute Check on Lesson 6-5 Transparency 6-6 Click the mouse button or press the Space Bar to display the answers. Find the perimeter of the given triangle. 1. ∆UVW, if ∆UVW ~ ∆UVW, MN = 6, NP = 8, MP = 12, and UW = ∆ABC, if ∆ABC ~ ∆DEF, BC = 4.5, EF = 9.9, and the perimeter of ∆DEF is Find x Find NO, if ∆MNO ~ ∆RSQ. Standardized Test Practice: ACBD x = x = 6 D x2x 9 8 x x – R Q S T O P N M

B C A K L J 3x + 1 5x A B C D F E x x Find x and the perimeter of DEF, if ∆DEF ~ ∆ABC Find x P S R T 8 x 6 x + 2 Find x if PT is an angle bisector A B C D E Find x, ED, and DB if ED = x – 3, CA = 20, EC = 16, and DB = x + 5 A B C D 6 4 P M N y 9 15 L Find y, if ∆ABC ~ ∆PNM Is CD a midsegment (connects two midpoints)?

B C A K L J 3x + 1 5x A B C D F E x x Find x P S R T 8 x 6 x + 2 Find x if PT is an angle bisector A B C D E Find x, ED, and DB if ED = x – 3, CA = 20, EC = 16, and DB = x + 5 A B C D 6 4 P M N y 9 15 L Find y, if ∆ABC ~ ∆PNM 3x = x – x + 12 = 40x – 8 20 = 4x 5 = x Find x and the perimeter of DEF, if ∆DEF ~ ∆ABC x = x = 96 x = 8 P = (x – 2) + x + 6 = 2x + 4 = 2(8) + 4 = 20 x – = ---- x x - 60 = 16x x = 140 x = 35 ED = 32 DB = = ---- x + 2 x 8x = 6x x = 12 x = = y 6y = 36 y = 6 Is CD a midsegment (connects two midpoints)? Since AC ≠ EC, then NO !

Summary & Homework Summary: –Similar triangles have perimeters proportional to the corresponding sides –Corresponding angle bisectors, medians, and altitudes of similar triangles have lengths in the same ratio as corresponding sides Homework: –Day 1: pg : 3-7, –Day 2: pg : 17-19, 22-24,